Internet Economicsכלכלת האינטרנט

Class 4 – Optimal Auctions

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Golden ballsLet’s warm up with some real-game theory:

Reality games and game theory…

• Scene 1• Scene 2

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Last week (1/4)• How to sell a single item to n bidders?

• Seller doesn’t know how much bidders are willing to pay– vi is the value of bidder i for the item.

• Getting this information via an auction.

• Game with incomplete information.

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Last week (2/4)

• Private value model: each person has a privately known value for the item.

• We saw: the two auctions are equivalent in the private value model.

• Auctions are efficient:dominant strategy for each player: truthfulness.

The English Auction:• Price starts at 0• Price increases until only one bidder is left.

Vickrey (2nd price) auction:• Bidders send bids.• Highest bid wins, pays 2nd highest bid.

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Last week (3/4)

• Dutch auctions and 1st price auctions are strategically equivalent. (asynchronous vs simple & fast)

• No dominant strategies. (tradeoff: chance of winning, payment upon winning.)

• Analysis in a Bayesian model:– Values are randomly drawn from a probability distribution.

• Strategy: a function. “What is my bid given my value?”

The Dutch Auction:• Price starts at max-price.• Price drops until a bidder agrees to buy.

1st-price auction:• Bidders send bids.• Highest bid wins, pays his bid.

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Last week (4/4)• We considered the simplest Bayesian model:

– 2 bidders.– Values drawn uniformly from [0,1].

Then:

In a 1st-price auction, it is a (Bayesian) Nash

equilibrium when all bidders bid

• An auction is efficient, if in (Bayesian) Nash equilibrium the bidder with the highest value always wins.– 1st price is efficient!

ivn

n 1

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Remark: Efficiency• We saw that both 2nd –price and 1st –price auctions

are efficient.

• What is efficiency (social welfare)?The total utility of the participants in the game (including the seller).

For each bidder: vi – pi

For the seller: (assuming it has 0 value for the item)

• Summing:

n

iip

1

n

ii

n

iii ppv

11

)(

n

iiv

1

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Optimal auctions• Usually the term optimal auctions stands for revenue

maximization.

• What is maximal revenue?– We can always charge the winner his value.

• Maximal revenue: optimal expected revenue in equilibrium.– Assuming a probability distribution on the values.– Over all the possible mechanisms.– Under individual-rationality constraints (later).

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Example: Spectrum auctions• One of the main triggers to

auction theory.

• FCC in the US sells spectrum, mainly for cellular networks.

• Improved auctions since the 90’s increased efficiency + revenue considerably.

• Complicated (“combinatorial”) auction, in many countries.– (more details further in the course)

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New Zealand Spectrum Auctions• A Vickrey (2nd price) auction was run in New Zealand to

sale a bunch of auctions. (In 1990)

• Winning bid: $100000Second highest: $6 (!!!!)Essentially zero revenue.

• NZ Returned to 1st price method the year after.– After that, went to a more complicated auction (in few weeks).

• Was it avoidable?

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1st or 2nd price?• Assume 2 bidders, uniform distribution on [0,1].

• Facts: (1) E[ max(v1,v2) ] = 2/3 (2) E[ min(v1,v2) ] = 1/3(in general, k’th highest value of n is (n+1-k)/n+1)

Revenue in 2nd price:• Bidders bid truthfully.• Revenue is 2nd highest bid.

Expected revenue = 1/3

Revenue in 1st price:• bidders bid vi/2.

• Revenue is the highest bid.

Expected revenue = E[ max(v1/2,v2/2) ]

= ½ E[ max(v1,v2)]

= ½ × 2/3 = 1/3 11

1

1E[revenue]

n

n

Revenue equivalence theorem• No coincidence!

– Somewhat unintuitively, revenue depends only on the way the winner is chosen, not on payments.

• Auction for a single good.• Values are independently drawn from distribution F (increasing)

Theorem (“revenue equivalence”):All auctions where:

1. the good is allocated to the bidder with the highest value2. Bidders can guarantee a utility of 0 by bidding 0.

yield the same revenue!

(more general: two auction with the same allocation rule yield the same revenue)

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Remark: Individual rationality• The following mechanism gains lots of revenue:

– Charge all players $10000000

• Bidder will clearly not participate.

• We thus have individual-rationality (or participation) constraints on mechanisms:bidders gain positive utility in equilibrium .– This is the reason for condition 2 in the theorem.

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All-pay auction (1/3)• Rules:

– Sealed bid– Highest bid wins– Everyone pay their bid

• Equilibrium with the uniform distribution:

b(v)=

• Does it achieve more or less revenue?– Note: Bidders shade their bids as the competition

increases.14

nvn

n 1

All-pay auction (2/3)• expected payment per each player: her bid.

• Each bidder bids • Expected payment for each bidder:

• Revenue: from n bidders

• Revenue equivalence!15

1

111

01

1111 1

0

1

0 n

n

nn

nv

n

ndvnv

n

ndvnv

n

n

1

1E[revenue]

n

n

nvn

nvb

1)(

All-pay auction (3/3)• Examples:

– crowdsourcing over the internet:• First person to complete a task for me gets a reward.• A group of people invest time in the task. (=payment)• Only the winner gets the reward.

– Advertising auction:• Collect suggestion for campaigns, choose a winner.• All advertiser incur cost of preparing the campaign.• Only one wins.

– Lobbying

– War of attrition• Animals invest (b1,b2) in fighting.

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What did we see so far• 2nd-price, 1st-price, all pay:

all obtain the same seller revenue.

• Revenue equivalence theorem:Auctions with the same allocation decisions earn the same expected seller revenue in equilibrium.– Constraint: individual rationality (participation constraint)

• Many assumptions:– statistical independence, – risk neutrality, – no externalities, – private values,– … 17

Next: Can we get better revenue?

• Can we achieve better revenue than the 2nd-price/1st price?

• If so, we must sacrifice efficiency. – All efficient auction have the same revenue….

• How?– Think about the New-Zealand case.

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Vickrey with Reserve Price• Seller publishes a minimum (“reserve”) price R.

• Each bidder writes his bid in a sealed envelope.

• The seller:– Collects bids– Open envelopes.

• Winner: Bidder with the highest bid, if bid is above R.

Otherwise, no one winsPayment: winner pays max{ 2nd highest bid, R}

Still Truthful? Yes. For bidders, exactly like an extra bidder bidding R.

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Can we get better revenue?• Let’s have another look at 2nd price auctions:

0 10

1

1 wins

2 wins

x

1 wins and pays x(his lowest winning

bid)x

v1

v2

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R

Can we get better revenue?• I will show that some reserve price improve revenue.

v10 1

0

1

v2 1 wins

2 wins

Revenue increased

Revenue increased

Revenue loss here (efficiency loss too)

R

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Can we get better revenue?

• Gain is at least 2R(1-R) R/2 = R2-R3

• Loss is at most R2 R = R3

0 10

1

1 wins

2 winsWe will be here with

probability R(1-R)

Average loss is R/2

When R2-2R3>0, reserve price of R is beneficial.(for example, R=1/4)

We will be here with

probability R2

Loss is always at

most R

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v1

v2

Reservation price• Can increase revenue!

• 2 bidders, uniform distribution: optimal reserve price = ½– Revenue: 5/12=0.412 > 1/3

• n bidders, uniform distribution: optimal reserve price = ½

Theorem: (Myerson ‘81)Vickrey auction with a reserve price maximizes revenue.– For a general family of distributions (uniform, exponential, normal,

and many others). – Reserve price is independent of n.

(Nobel prize, 2007)

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Reservation priceLet’s see another example:

How do you sell one item to one bidder?– Assume his value is drawn uniformly from [0,1].

• Optimal way: reserve price. – Take-it-or-leave-it-offer.

• Let’s find the optimal reserve price:E[revenue] = ( 1-F(R) ) × R = (1-R) ×R

R=1/2

• Surprising? No. We said that the optimal reserve price does not depend on n.

021)1(

RR

RR

Probability that the buyer will

accept the priceThe payment for

the seller

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Back to New Zealand• Recall:

Vickrey auction.Highest bid: $100000. Revenue: $6.

• Two things to learn:– Seller can never get the whole pie.

• “information rent” for the buyers.

– Reserve price can help.• But what if R=$50000 and highest bid was $45000?

• Of the unattractive properties of Vickrey Auctions:– Low revenue despite high bids.– 1st-price may earn same revenue, but no explanation needed…

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Summary: Efficiency vs. revenuePositive or negative correlation ?

• Always: Revenue ≤ efficiency– Due to Individual rationality.More efficiency makes the pie larger!

• However, for optimal revenue one needs to sacrifice some efficiency.

• Consider two competing sellers: one optimizing revenue the other optimizing efficiency.– Who will have a higher market share?– In the longer terms, two objectives are combined. 28

Next week

• Designing dominant-strategy mechanisms for more general environments.– the magic of the VCG mechanism.

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